I'd suggest adding Client_ID to all of your tables. So instead Order_ID being the primary key for Orders table and foreign key on Order_Items tables, it would now be a composite key of Client_ID, Order_ID. The primary advantage of this is that the existing key values don't change, which will retain transactional history. Also, since consolidating data across clients is being done in the physical model for maintainability reasons (the client entities themselves are not merging in the real world), the case usage and queries of the application will tend to be client specific.

For example, client ABC, or perhaps the sales representabive for client ABC, will login to the application and want to see sales transactions only for ABC. Having Client_ID in your transactional tables will help facilitate this.